%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% amlt.sty: LaTeX Style file for Articles to be published in
%%
%% Annals of Mathematics
%% 
%% Princeton University and the
%% Institute for Advanced Study
%%
%% Published by Princeton University Press
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Macros written by Amy Hendrickson
%% TeXnology Inc, Brookline, MA
%% 617 738-8029, amyh@ai.mit.edu
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\spaces{\space\space\space\space\space\space\space\space\space\space}
\def\spacess{\message{\spaces\spaces\spaces\spaces\spaces\spaces\spaces}}
\spacess
\spacess
\message{Annals of Mathematics Style: Current Version: 1.0. March 25, 1992}
\spacess
\spacess

%% Adaptation of LaTeX book.sty

\def\@ptsize{0} \@namedef{ds@11pt}{\def\@ptsize{1}}
\@namedef{ds@12pt}{\def\@ptsize{2}} 
\@twosidetrue \@mparswitchtrue \def\ds@draft{\overfullrule 5pt} 
\@options

%======= Variations on bk11.sty

\lineskip 1pt \normallineskip 1pt
\def\baselinestretch{1}

\def\@normalsize{\@setsize\normalsize{13.95pt}\xipt\@xipt
\abovedisplayskip=6.95pt plus3.5pt minus 3pt
\belowdisplayskip=\abovedisplayskip
%\abovedisplayskip 11pt plus3pt minus6pt\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus3pt\belowdisplayshortskip 6.5pt plus3.5pt
minus3pt\let\@listi\@listI} 
\def\small{\@setsize\small{12pt}\xpt\@xpt
\abovedisplayskip 10pt plus2pt minus5pt\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus3pt\belowdisplayshortskip 6pt plus3pt minus3pt
\def\@listi{\leftmargin\leftmargini \topsep 6pt plus 2pt minus 2pt\parsep 3pt
plus 2pt minus 1pt
\itemsep \parsep}}
\def\footnotesize{\@setsize\footnotesize{11pt}\viiipt\@viiipt
\abovedisplayskip 8pt plus 2pt minus 4pt\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus 1pt\belowdisplayshortskip 4pt plus 2pt minus
2pt
\def\@listi{\leftmargin\leftmargini \topsep 4pt plus 2pt minus 2pt\parsep 2pt
plus 1pt minus 1pt
\itemsep \parsep}}
\def\scriptsize{\@setsize\scriptsize{9.5pt}\viiipt\@viiipt}
\def\tiny{\@setsize\tiny{7pt}\vipt\@vipt}
\def\large{\@setsize\large{14pt}\xiipt\@xiipt}
\def\Large{\@setsize\Large{18pt}\xivpt\@xivpt}
\def\LARGE{\@setsize\LARGE{22pt}\xviipt\@xviipt}
\def\huge{\@setsize\huge{25pt}\xxpt\@xxpt}
\def\Huge{\@setsize\Huge{30pt}\xxvpt\@xxvpt}
\normalsize 

\oddsidemargin .25in \evensidemargin 1.25in 
\marginparwidth 1in \marginparsep 7pt 
 \topmargin .73in \headheight 12pt 
 \footskip .38in 

 \columnsep 10pt \columnseprule 0pt 

\footnotesep 7.7pt 
\skip\footins 10pt plus 4pt minus 2pt 
\floatsep 12pt plus 2pt minus 2pt \textfloatsep 20pt plus 2pt minus 4pt
\intextsep 12pt plus 2pt minus 2pt \@maxsep 20pt \dblfloatsep 12pt plus 2pt
minus 2pt \dbltextfloatsep 20pt plus 2pt minus 4pt \@dblmaxsep 20pt 
\@fptop 0pt plus 1fil \@fpsep 8pt plus 2fil \@fpbot 0pt plus 1fil 
\@dblfptop 0pt plus 1fil \@dblfpsep 8pt plus 2fil \@dblfpbot 0pt plus 1fil
\marginparpush 5pt 

\parskip 0pt plus 1pt  \partopsep 3pt plus 1pt minus 2pt 
\@lowpenalty 51 \@medpenalty 151 \@highpenalty 301 
\@beginparpenalty -\@lowpenalty \@endparpenalty -\@lowpenalty \@itempenalty
-\@lowpenalty 


\leftmargini 2.5em
\leftmarginii 2.2em \leftmarginiii 1.87em \leftmarginiv 1.7em \leftmarginv 1em
\leftmarginvi 1em
\leftmargin\leftmargini
\labelsep .5em
\labelwidth\leftmargini\advance\labelwidth-\labelsep
\def\@listI{\leftmargin\leftmargini \parsep 4.5pt plus 2pt minus 1pt\topsep
9pt plus 3pt minus 5pt\itemsep 4.5pt plus 2pt minus 1pt}
\let\@listi\@listI
\@listi 
\def\@listii{\leftmargin\leftmarginii
 \labelwidth\leftmarginii\advance\labelwidth-\labelsep
 \topsep 4.5pt plus 2pt minus 1pt
 \parsep 2pt plus 1pt minus 1pt
 \itemsep \parsep}
\def\@listiii{\leftmargin\leftmarginiii
 \labelwidth\leftmarginiii\advance\labelwidth-\labelsep
 \topsep 2pt plus 1pt minus 1pt 
 \parsep \z@ \partopsep 1pt plus 0pt minus 1pt
 \itemsep \topsep}
\def\@listiv{\leftmargin\leftmarginiv
 \labelwidth\leftmarginiv\advance\labelwidth-\labelsep}
\def\@listv{\leftmargin\leftmarginv
 \labelwidth\leftmarginv\advance\labelwidth-\labelsep}
\def\@listvi{\leftmargin\leftmarginvi
 \labelwidth\leftmarginvi\advance\labelwidth-\labelsep}

%%%%%%%% end of bk11.sty variation

\def\labelenumi{\arabic{enumi}.} 
\def\theenumi{\arabic{enumi}} 
\def\labelenumii{(\alph{enumii})}
\def\theenumii{\alph{enumii}}
\def\p@enumii{\theenumi}
\def\labelenumiii{\roman{enumiii}.}
\def\theenumiii{\roman{enumiii}}
\def\p@enumiii{\theenumi(\theenumii)}
\def\labelenumiv{\Alph{enumiv}.}
\def\theenumiv{\Alph{enumiv}} 
\def\p@enumiv{\p@enumiii\theenumiii}

\def\labelitemi{$\bullet$}
\def\labelitemii{\bf --}
\def\labelitemiii{$\ast$}
\def\labelitemiv{$\cdot$}

\def\verse{\let\\=\@centercr 
 \list{}{\itemsep\z@ \itemindent -1.5em\listparindent \itemindent 
 \rightmargin\leftmargin\advance\leftmargin 1.5em}\item[]}
\let\endverse\endlist
\def\quotation{\list{}{\listparindent 1.5em
 \itemindent\listparindent
 \rightmargin\leftmargin\parsep 0pt plus 1pt}\item[]}
\let\endquotation=\endlist
\def\quote{\list{}{\rightmargin\leftmargin}\item[]}
\let\endquote=\endlist

\def\descriptionlabel#1{\hspace\labelsep \bf #1}
\def\description{\list{}{\labelwidth\z@ \itemindent-\leftmargin
 \let\makelabel\descriptionlabel}}
\let\enddescription\endlist
\newdimen\descriptionmargin
\descriptionmargin=3em


\arraycolsep 5pt \tabcolsep 6pt \arrayrulewidth .4pt \doublerulesep 2pt 
\tabbingsep \labelsep 
\skip\@mpfootins = \skip\footins
\fboxsep = 3pt \fboxrule = .4pt 


\def\@pnumwidth{1.55em}

\setcounter{topnumber}{2}
\def\topfraction{.7}
\setcounter{bottomnumber}{1}
\def\bottomfraction{.3}
\setcounter{totalnumber}{3}
\def\textfraction{.2}
\def\floatpagefraction{.5}
\setcounter{dbltopnumber}{2}
\def\dbltopfraction{.7}
\def\dblfloatpagefraction{.5}

\long\def\@makecaption#1#2{\vskip 10pt 
\vtop{\hbox{\eightpoint\bf#1}\raggedright\parskip=6pt\relax\eightpoint
\hsize=26pc\vskip-6pt\noindent#2}}

\newcounter{part}
\newcounter {chapter}
\newcounter {section}[chapter]
\newcounter {subsection}[section]
\newcounter {subsubsection}[subsection]
\newcounter {paragraph}[subsubsection]
\newcounter {subparagraph}[paragraph]

\newcounter{figure}[chapter]
\def\thefigure{\thechapter.\@arabic\c@figure}
\def\thesection{\arabic{section}}
\def\fps@figure{tbp}
\def\ftype@figure{1}
\def\ext@figure{lof}
\def\fnum@figure{Figure \thefigure}
\def\figure{\@float{figure}}
\let\endfigure\end@float
\@namedef{figure*}{\@dblfloat{figure}}
\@namedef{endfigure*}{\end@dblfloat}
\newcounter{table}[chapter]
\def\thetable{\thechapter.\@arabic\c@table}
\def\fps@table{tbp}
\def\ftype@table{2}
\def\ext@table{lot}
\def\fnum@table{Table \thetable}
\def\endtable{\end@float}
\@namedef{table*}{\@dblfloat{table}}
\@namedef{endtable*}{\end@dblfloat}

\mark{{}{}}  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%% Annals of Mathematics Modifications %%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% This file includes: 
%% 1) Font declarations, 
%% 2) Page set up, 
%% 3) Title page 
%% 4) Section heads,
%% 5) Equation macros, autonumbering equations, etc.,
%% 6) Figure and Table Captions,
%% 7) End matter macros: Bibliography, Appendix, etc.,
%% 8) Footnotes,
%% 9) Theorem type environments
%% 10) Listing
%% 11) Article and journal Table of Contents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 1) Font declarations, 
% Computer Modern fonts 

% Small Caps
\font\elevensc=cmcsc10 scaled\magstephalf
\font\tensc=cmcsc10
\font\ninesc=cmcsc10 scaled900
\font\eightsc=cmcsc10 scaled800

\font\twelverm=cmr12
\font\elevenrm=cmr10 scaled \magstephalf
\font\ninerm=cmr9
\font\eightrm=cmr8
\font\sixrm=cmr6
\font\fiverm=cmr5

\font\eleveni=cmmi10 scaled\magstephalf
\font\ninei=cmmi9
\font\eighti=cmmi8
\font\sixi=cmmi6
\font\fivei=cmmi5
\skewchar\ninei='177 \skewchar\eighti='177 \skewchar\sixi='177
\skewchar\eleveni='177

\font\elevensy=cmsy10 scaled\magstephalf
\font\ninesy=cmsy9
\font\eightsy=cmsy8
\font\sixsy=cmsy6
\font\fivesy=cmsy5
\skewchar\ninesy='60 \skewchar\eightsy='60 \skewchar\sixsy='60
\skewchar\elevensy='60

\font\eighteenbf=cmbx10 scaled\magstep3

\font\twelvebf=cmbx10 scaled \magstep1
\font\elevenbf=cmbx10 scaled \magstephalf
\font\tenbf=cmbx10
\font\ninebf=cmbx9
\font\eightbf=cmbx8
\font\sixbf=cmbx6
\font\fivebf=cmbx5

\font\elevenit=cmti10 scaled\magstephalf
\font\nineit=cmti9
\font\eightit=cmti8

% Fonts for bold math
\font\eighteenmib=cmmib10 scaled \magstep3
\font\twelvemib=cmmib10 scaled \magstep1
\font\elevenmib=cmmib10 scaled\magstephalf
\font\tenmib=cmmib10
\font\eightmib=cmmib10 scaled 800 
\font\sixmib=cmmib10 scaled 600

\font\eighteensyb=cmbsy10 scaled \magstep3 
\font\twelvesyb=cmbsy10 scaled \magstep1
\font\elevensyb=cmbsy10 scaled \magstephalf
\font\tensyb=cmbsy10 
\font\eightsyb=cmbsy10 scaled 800
\font\sixsyb=cmbsy10 scaled 600
 
\font\tenex=cmex10     
\font\elevenex=cmex10 scaled\magstephalf    


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Font families

\def\elevenpoint{\def\rm{\fam0\elevenrm}%
  \textfont0=\elevenrm \scriptfont0=\eightrm \scriptscriptfont0=\sixrm
  \textfont1=\eleveni \scriptfont1=\eighti \scriptscriptfont1=\sixi
  \textfont2=\elevensy \scriptfont2=\eightsy \scriptscriptfont2=\sixsy
  \textfont3=\elevenex \scriptfont3=\elevenex \scriptscriptfont3=\elevenex
  \def\bf{\fam\bffam\elevenbf}%
  \def\it{\fam\itfam\elevenit}%
  \textfont\bffam=\elevenbf \scriptfont\bffam=\eightbf
   \scriptscriptfont\bffam=\sixbf
\normalbaselineskip=13.95pt
  \setbox\strutbox=\hbox{\vrule height9.5pt depth4.4pt width0pt\relax}%
  \normalbaselines\rm}

\elevenpoint % default fonts and baselineskip

\def\ninepoint{\def\rm{\fam0\ninerm}%
  \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
  \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
  \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
  \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
  \def\it{\fam\itfam\nineit}%
  \textfont\itfam=\nineit
  \def\bf{\fam\bffam\ninebf}%
  \textfont\bffam=\ninebf \scriptfont\bffam=\sixbf
   \scriptscriptfont\bffam=\fivebf
\normalbaselineskip=11pt
  \setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt\relax}%
  \normalbaselines\rm}

\def\eightpoint{\def\rm{\fam0\eightrm}%
  \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
  \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei
  \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
  \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
  \def\it{\fam\itfam\eightit}%
  \textfont\itfam=\eightit
  \def\bf{\fam\bffam\eightbf}%
  \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf
   \scriptscriptfont\bffam=\fivebf
\normalbaselineskip=12pt
  \setbox\strutbox=\hbox{\vrule height8.5pt depth3.5pt width0pt\relax}%
  \normalbaselines\rm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Font families for bold math in title and section heads

\def\eighteenbold{\def\rm{\fam0\eighteenbf}%
  \textfont0=\eighteenbf \scriptfont0=\twelvebf \scriptscriptfont0=\tenbf
  \textfont1=\eighteenmib \scriptfont1=\twelvemib\scriptscriptfont1=\tenmib
  \textfont2=\eighteensyb \scriptfont2=\twelvesyb\scriptscriptfont2=\tensyb
  \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
   % should be eighteenex but we are running out of font space
  \def\bf{\fam\bffam\eighteenbf}%
  \textfont\bffam=\eighteenbf \scriptfont\bffam=\twelvebf
   \scriptscriptfont\bffam=\tenbf
\normalbaselineskip=20pt
  \setbox\strutbox=\hbox{\vrule height13.5pt depth6.5pt width0pt\relax}%
\everymath {\fam0 }
\everydisplay {\fam0 }
  \normalbaselines\rm}

\def\elevenbold{\def\rm{\fam0\elevenbf}%
  \textfont0=\elevenbf \scriptfont0=\eightbf \scriptscriptfont0=\sixbf%
  \textfont1=\elevenmib \scriptfont1=\eightmib \scriptscriptfont1=\sixmib%
  \textfont2=\elevensyb \scriptfont2=\eightsyb \scriptscriptfont2=\sixsyb%
  \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex%
  \def\bf{\fam\bffam\elevenbf}%
  \textfont\bffam=\elevenbf \scriptfont\bffam=\eightbf%
   \scriptscriptfont\bffam=\sixbf%
\normalbaselineskip=14pt%
  \setbox\strutbox=\hbox{\vrule height10pt depth4pt width0pt\relax}%
\everymath {\fam0 }%
\everydisplay {\fam0 }%
  \normalbaselines\bf}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 2) Page set up

\textwidth=31pc

\headsep16pt

\textheight=48.5pc

\parindent=22pt
\parskip=0pt

\widowpenalty=10000
\clubpenalty=10000


\skip\footins=20pt
\dimen\footins=3in % maximum footnote height

\abovedisplayskip=6.95pt plus3.5pt minus 3pt
\belowdisplayskip=\abovedisplayskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 3) Title page 

 %#1= Volume number, #2=year of publication
\def\annalsline#1#2{\vfill\eject
\ifodd\c@page\else % first page of article on right.
\line{\hfill}
\vfill\eject\fi
\global\titletrue
\def\currannalsline{\eightrm Annals of Mathematics,
{\eightbf#1} (#2), \thepages}}

\def\titleheadline#1{\def\one{#1}\ifx\one\empty\else
\def\thetitle{{\frenchspacing%
\let\\ \relax\eightsc\uppercase{#1}}}\fi}

\newif\ifshort
\def\shortname#1{\global\shorttrue\xdef\theauthors{{\eightsc\uppercase{#1}}}}
\let\shorttitle\titleheadline

\def\yearofpublication#1{\def\theyear{#1}}
\def\onpages#1#2{\def\thepages{#1--#2}}

\def\thismuchskip[#1]{\vskip#1pt}
\def\ilook{\ifx\next[ \let\go\thismuchskip\else
\let\go\relax\vskip1pt\fi\go}

\def\institution#1{\def\theinstitutions{\vbox{\baselineskip10pt
\def\and{{\eightrm and }}
\def\\{\futurelet\next\ilook}\eightsc #1}}}
\let\institutions\institution



\def\startingpage#1{\def\one{#1}\ifx\one\empty\global\c@page=1\else
\global\c@page=#1\fi
\theoremcount=0 \eqcount=0 \sectioncount=0 
\onpages{\def\one{#1}\ifx\one\empty???\else#1\fi}
{\expandafter\ifx\csname lastpage\endcsname\relax ???%
\else\lastpage\fi}}

\def\endarticle{\ifRefsUsed\else\vskip21pt\theinstitutions
\nobreak\vskip8pt\vbox{\thereceived\therevised}\fi
\write\@auxout{\string\gdef\string\lastpage{\the\c@page}}}

\let\saveenddoc\enddocument

\def\enddocument{\endarticle\saveenddoc}


\long\def\x@makefntext{\noindent\hbox to22pt{\hfill$^*$}%
\eightpoint\normalbaselineskip11pt\normalbaselines}

\long\def\xftnttext#1{\insert\footins{\eightpoint
    \interlinepenalty\interfootnotelinepenalty 
    \splittopskip\footnotesep
    \splitmaxdepth \dp\strutbox \floatingpenalty \@MM
    \hsize\columnwidth \@parboxrestore
\x@makefntext{\rule{\z@}{\footnotesep}\ignorespaces
      #1\strut}}}

\newif\ifacks
\long\def\acknowledgements#1{\def\one{#1}\ifx\one\empty\else%
\global\ackstrue\xftnttext{#1}\fi}
\let\acknowledgement\acknowledgements

\def\received#1{\def\thereceived{\vskip1sp\centerline{\ninerm Received #1.}}}

\def\revised#1{\def\therevised{\vskip-2.9pt 
\centerline{\ninerm Revised #1.}}}

\def\title#1{\titleheadline{#1}\vbox to87pt{\vfill
\baselineskip=18pt
\parindent=0pt
\overfullrule=0pt
\hyphenpenalty=10000
\everypar={\hskip\parfillskip\relax}
\hbadness=10000
\def\\ {\vskip1sp}
\eighteenbold#1\vskip1sp}
}

\newif\ifauthor
\def\author#1{\vskip15pt
\hbox to\hsize{\hss\tenrm By \tensc#1\ifacks\global\acksfalse*\fi\hss}
\ifshort\else\xdef\theauthors{{\eightsc\uppercase{#1}}}\fi%
\vskip7pt
\vskip\baselineskip
\global\authortrue\everypar={\global\authorfalse\everypar={}}}

\def\twoauthors#1#2{\vskip15pt
\hbox to\textwidth{\hss%
\tenrm By \tensc#1 {\tenrm and} #2\ifacks\global\acksfalse*\fi\hss}
\ifshort\else\xdef\theauthors{{\eightsc\uppercase{#1 and #2}}}\fi
\vskip7pt
\vskip\baselineskip
\global\authortrue\everypar={\global\authorfalse\everypar={}}}


\let\authors\author

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 4) Section heads, counters

\newcount\theoremcount
\newcount\sectioncount
\newcount\eqcount

\newif\ifspecialnumon
\def\specialnumber#1{\global\specialnumontrue\gdef\spnum{\hbox{$\rm#1$}}}

\let\spnumber\specialnumber

\def\eqnumber=#1 {\global\eqcount=#1 \global\advance\eqcount by-1\relax}
\def\sectionnumber=#1 {\global\sectioncount=#1 
\global\advance\sectioncount by-1\relax}
\def\proclaimnumber=#1 {\global\theoremcount=#1 
\global\advance\theoremcount by-1\relax}

\newif\ifsection
\newif\ifsubsection

\def\intro{\global\authorfalse%
\hbox to\textwidth{\hss\bf Introduction\hss}\everypar={}\vskip6pt}

\def\elevenboldmath#1{$#1$\egroup}
\def\mathbold{\hbox\bgroup\protect\elevenbold\elevenboldmath}

\def\section#1{\global\theoremcount=0
\global\eqcount=0
\ifauthor\global\authorfalse\else
\vskip18pt plus 18pt\fi
{\parindent=0pt
\everypar={\hskip\parfillskip}
\def\\ {\vskip1sp}\elevenpoint\bf%
\ifspecialnumon\global\specialnumonfalse\spnum%
\gdef\sectnum{\spnum}%
\else\interlinepenalty=10000%
\global\advance\sectioncount by1\relax\the\sectioncount%
\gdef\sectnum{\the\sectioncount}%
\fi.\hskip6pt#1\vrule depth 12pt width0pt}\hskip\parfillskip%
\break\global\sectiontrue%
\global\sectiontrue%
\xdef\@currentlabel{\the\sectioncount}%
\everypar={\global\sectionfalse\global\interlinepenalty=0\everypar={}}%
\ignorespaces}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 5) Equation macros

\def\refstepcounter#1{}
\newif\ifinmath %% Used when tables that don't have extra space 
                %% on outside edge are including.

%automatic left side equation numbers: 
% **** \endequation is redefined below ****
\def\endequation{\ifsplitmath$$\global\splitmathfalse\else
\leqno\ifmathqed\hbox to0pt{\hbox to\textwidth{\hfill\qed}\hss}\fi%
\Eeqnu% $$ BRACE MATCHING HACK
$$\fi\global\@ignoretrue}

\def\@eqnnum{\hbox to .01pt{}%
\rlap{\rm \hskip -\displaywidth\Eeqnu}}

%% Splitmath, to break equations into two lines easily.
%% To use: \begin{splitmath} xyz\\ 123\\ \end{splitmath}
%% Will produce equation numbers when used with
%% \begin{equation}...\end{equation}, i.e.,
%% \begin{equation}
%% \begin{splitmath} xyz\\ 123\\ \end{splitmath}
%% \end{equation}

%% Vertically centers equation number.

\newif\ifsplitmath
\def\splitmath#1\\#2\\{\ifmmode
\ifinequation\global\splitmathtrue\fi
\setbox1\vbox{\hbox to\linewidth{\setbox0=\hbox{\rm(\theequation)\hskip1em}%
\hskip\wd0$\displaystyle#1$\hfill}\vskip1pt
\global\advance\eqcount by-1 \relax%
\hbox to\linewidth{\hfill$\displaystyle#2$\ifmathqed\qquad\qed\fi}%
}%
\vcenter{\gdef\@currentlabel{\theequation}%
\copy1%
\dimen0=\ht1 \advance\dimen0 by\dp1
\vskip-\dimen0
\vbox to\dimen0{\hrule\vfill%
\hbox{\rm\theequation}%
\vfill\hrule}%
}
\else
\vskip-\lastskip
\vskip\abovedisplayskip
\hbox to\linewidth{$\displaystyle#1$\hfill}%
\vskip1pt
\hbox to\linewidth{\hfill$\displaystyle#2$\ifmathqed\qquad\qed\fi}%
\vskip\belowdisplayskip\noindent\ignorespaces\fi}

%% Math, delay labelling for cross-referencing

\def\eqnarray{\stepcounter{equation}\let\@currentlabel=\theequation
\global\@eqnswtrue
\global\@eqcnt\z@\tabskip\@centering\let\\=\@eqncr
$$\halign to \displaywidth\bgroup\@eqnsel\hskip\@centering
  $\displaystyle\tabskip\z@{##}$&\global\@eqcnt\@ne 
\hskip1em%  \hskip 2\arraycolsep <==== changed at Fred's request
\hfil${##}$\hfil
&\global\@eqcnt\tw@ 
\hskip1em% \hskip 2\arraycolsep <==== changed out at Fred's request
$\displaystyle\tabskip\z@{##}$\hfil%
 \tabskip\@centering&\llap{##}\tabskip\z@\cr}

\def\endeqnarray{\@@eqncr\egroup
      \global\advance\c@equation\m@ne$$\global\@ignoretrue}

\let\saveeqnarray\eqnarray
\let\saveendeqnarray\endeqnarray

\def\eqnarray{\def\label##1{\gdef\catchlabel{##1}}\saveeqnarray}
\def\endeqnarray{\saveendeqnarray\gdef\catchlabel{}}

\def\[{\relax \ifmmode \@badmath \else \ifvmode \nointerlineskip 
\makebox [.6\linewidth ]\fi $$\fi}

\def\saveqed{\saveeqno\qed}
\def\]{\relax\ifmmode\ifinner\@badmath\else
\ifmathqed\let\go\saveqed\else\let\go\relax
\fi\go$$\fi\else\@badmath\fi\ignorespaces}

\let\savebracket\[
\def\[{\relax\global\inmathtrue\savebracket}

\let\saveendbracket\]
\def\]{\relax\global\inmathtrue\saveendbracket}

\let\saveequation\equation
\let\savelabel\label

\def\catchlabel{}

\newif\ifinequation
\def\equation{\global\inequationtrue\def\label##1{\gdef\catchlabel{##1}}%
\global\inmathtrue\saveequation
\let\\ \break\hfill}

\let\saveendequation\endequation
\def\endequation{\saveendequation
\ifx\catchlabel\empty\else
\savelabel{\catchlabel}%
\gdef\catchlabel{}\fi
\global\inequationfalse\global\inmathfalse}

\def\theequation{\Eeqnu}

\newif\ifspequation
\def\Xspeqnu#1{\global\spequationtrue
\global\specialnumontrue\gdef\spnum{\hbox{$\rm#1$}}}

\def\speqnu#1{\specialnumber{$\rm#1$}}

\let\saveeqno\eqno
\let\eqno\leqno %automatic left side equation numbers

\newif\ifineqalignno
\let\saveleqalignno\leqalignno
\def\leqalignno{\let\eqnu\Eeqnu\saveleqalignno}

\let\eqalignno\leqalignno

\def\sectandeqnum{%
\ifspecialnumon\global\specialnumonfalse
\spnum\gdef\eqnum{\spnum}\else\global\firstlettertrue
\global\advance\eqcount by1 
\ifappend\applett\else\the\sectioncount\fi.%
\the\eqcount%
\xdef\eqnum{\ifappend\applett\else\the\sectioncount\fi.\the\eqcount}\fi
\xdef\@currentlabel{\eqnum}}

\def\eqnu{\leqno{\hbox{\elevenrm\ifspequation\else(\fi\sectandeqnum
\ifspequation\global\spequationfalse\else)\fi}}}

\def\Speqnu{\global\setbox\leqnobox=\hbox{\elevenrm
\ifspequation\else%
(\fi\sectandeqnum\ifspequation\global\spequationfalse\else)\fi}}

\def\Eeqnu{\hbox{\elevenrm\ifspequation\else%
(\fi\sectandeqnum\ifspequation\global\spequationfalse\else)\fi}%
\ifx\catchlabel\empty\else
\savelabel{\catchlabel}\ifsplitmath\else\gdef\catchlabel{}\fi\fi}

\newif\iffirstletter
\global\firstlettertrue

\def\eqletter#1{\global\specialnumontrue\iffirstletter\global\firstletterfalse
\global\advance\eqcount by1\fi%
\gdef\spnum{\ifappend\applett\else%
\the\sectioncount\fi.\the\eqcount\hbox{$\rm#1$}}}


%% Proof box to be used when proof ends with equation.

\newif\ifmathqed
\def\inmathqed{\global\mathqedtrue}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \mtable, Math table to make binary table easily

%% Use:
% \mtable
% &n_1&n_2&n_3&n_4&n_5&n_6\cr
% \Delta_1&M_3&M_2&0&0&0&0\cr
% \Delta_2&0&0&M_1&M_3&0&0\cr
% \endmtable

\newcount\linenum
\newcount\colnum
\def\spline{\omit&\multispan{\the\colnum}{\hrulefill}\cr}
\def\colcounter{\ifnum\linenum=1\global\advance\colnum by1\fi}

\def\everyline{\noalign{\global\advance\linenum by1\relax}%
\ifnum\linenum=2\spline\fi}

\def\mtable{\bgroup\offinterlineskip
\everycr={\everyline}\global\linenum=0
\halign\bgroup\vrule height 10pt depth 4pt width0pt
\hfill$##$\hfill\hskip6pt\ifnum\linenum>1
\vrule\fi&&\colcounter\hskip12pt\hfill$##$\hfill\hskip12pt\cr}

\def\endmtable{\crcr\egroup\egroup}


\def\big#1{{\hbox{$\left#1\vbox to 10pt{}\right.\n@space$}}}
\def\Big#1{{\hbox{$\left#1\vbox to 13pt{}\right.\n@space$}}}
\def\bigg#1{{\hbox{$\left#1\vbox to 16pt{}\right.\n@space$}}}
\def\Bigg#1{{\hbox{$\left#1\vbox to 19pt{}\right.\n@space$}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 6) Figure and Table Captions.

\def\fig#1#2{\iftitle\begin{figure}[h]\else\begin{figure}[t]\fi
\vbox to#2{\vfill}\vskip1sp
\centerline{\eightsc Figure #1}
\vskip2pt
\end{figure}
}

\def\figcaption#1#2#3{\iftitle\begin{figure}[h]\else\begin{figure}[t]\fi
\vbox to#3{\vfill}\vskip1sp
\setbox0=\hbox{\eightsc Figure #1.\hskip12pt\eightpoint #2}
\ifdim\wd0>\hsize
\noindent\eightsc Figure #1.\hskip12pt\eightpoint #2
\else
\centerline{\eightsc Figure #1.\hskip12pt\eightpoint #2}
\fi
\end{figure}}

\def\wfigcaption#1#2#3#4{\iftitle\begin{figure}[h]\else\begin{figure}[t]\fi
\vskip4pt %<===topadjust to match height of ascenders on opposing page.
\hbox to\hsize{\hss\vbox{\hrule height .25pt width #4
\hbox to #4{\vrule width .25pt height #3\hfill\vrule width .25pt height #3}
\hrule height.25pt}\hss}
\vskip1sp
\setbox0=\hbox{\eightsc Figure #1.\hskip12pt\eightpoint #2}
\ifdim\wd0>\hsize
\noindent\eightsc Figure #1.\hskip12pt\eightpoint #2
\else
\centerline{\eightsc Figure #1.\hskip12pt\eightpoint #2}
\fi
\end{figure}}

\def\wfig#1#2#3{\iftitle\begin{figure}[h]\else\begin{figure}[t]\fi
\vskip4pt %<===topadjust to match height of ascenders on opposing page.
\hbox to\hsize{\hss\vbox{\hrule height .25pt width #3
\hbox to #3{\vrule width .25pt height #2\hfill\vrule width .25pt height #2}
\hrule height.25pt}\hss}
\vskip1sp
\centerline{\eightsc Figure #1}
\vskip2pt
\end{figure}
}

\def\tablehead#1{\vskip6pt
\centerline{\eightsc Table #1}
\vskip6pt}

\def\tabcaption#1#2{\vskip6pt
\setbox0=\hbox{\eightsc Table #1.\hskip12pt\eightpoint #2}
\ifdim\wd0>\hsize
\noindent\eightsc Table #1.\hskip12pt\eightpoint #2
\else
\centerline{\eightsc Table #1.\hskip12pt\eightpoint #2}
\fi
\vskip6pt}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 7) End Matter

\newif\ifappend

\def\appendix#1#2{\def\applett{#1}\def\two{#2}%
\global\appendtrue
\global\theoremcount=0
\global\eqcount=0
\vskip18pt plus 18pt
\vbox{\parindent=0pt
\everypar={\hskip\parfillskip}
\def\\ {\vskip1sp}\elevenbold Appendix% 
\ifx\applett\empty\gdef\applett{A}\ifx\two\empty\else.\fi%
\else\ #1.\fi\hskip6pt#2\vskip12pt}%
\global\sectiontrue%
\everypar={\global\sectionfalse\everypar={}}\nobreak\ignorespaces}


%%% Bibliography

\def\bibline{\hbox to30pt{\hrulefill}\/\/}

\def\name#1{{\eightsc#1}}

\bibliographystyle{plain}

\newcount\bibnum


\def\bibitem#1{\global\advance\bibnum by1 %
\immediate\write\@auxout{\string\bibcite{#1}{\the\bibnum}}%
\leavevmode\hbox to18pt{\hbox to13.2pt{\hss[\the\bibnum]}\hfill}\ignorespaces}


\newif\ifRefsUsed
\long\def\references{\global\RefsUsedtrue\vskip21pt
\theinstitutions
\everypar={}\global\bibnum=0
\vskip20pt\goodbreak\bgroup
\vbox{\centerline{\eightsc References}\vskip6pt}%
\ifdim\maxbibwidth>0pt
\leftskip=\maxbibwidth%
\parindent=-\maxbibwidth%
\else
\leftskip=18pt%
\parindent=-18pt%
\fi
\ninepoint
\frenchspacing
\nobreak\ignorespaces}

\def\endreferences{\vskip1pt\egroup\everypar={}%
\nobreak\vskip8pt\vbox{\thereceived\therevised}}

\newdimen\maxbibwidth
\def\AuthorRefNames [#1] {\def\bibitem{\spbibitem}
\setbox0=\hbox{[#1] }\global\maxbibwidth=\wd0\relax}

\def\spbibitem#1{\global\advance\bibnum by1 %
\immediate\write\@auxout{\string\bibcite{#1}{#1}}%
\leavevmode\hbox to\maxbibwidth{\hss[#1]\hfill}\ignorespaces}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 8) Footnotes

\def\footnoterule{\kern-3pt\hrule width1in height.5pt\kern2.5pt}

\long\def\@makefntext#1{{\eightpoint\normalbaselineskip11pt
\normalbaselines\parindent 1em\noindent 
\hbox to 22pt{\hss$^{\@thefnmark}$}#1}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 9) Theorem type environments

\newif\iffirstadded
\firstaddedtrue
\newif\ifadded
\def\addletter#1{\global\addedtrue\global\specialnumontrue\def\addedlett{#1}}
\def\addedlett{}

\def\alltheoremnums{%
\ifspecialnumon\global\specialnumonfalse
\ifadded\global\addedfalse
\iffirstadded\global\firstaddedfalse
\global\advance\theoremcount by1 \fi
\ifappend\applett\else\the\sectioncount\fi.\the\theoremcount\addedlett%
\xdef\theoremnum{\ifappend\applett\else\the\sectioncount\fi.%
\the\theoremcount\addedlett}%
\else$\rm\spnum$\def\theoremnum{{$\rm\spnum$}}\fi%
\else\global\firstaddedtrue
\global\advance\theoremcount by1 
\ifappend\applett\else\the\sectioncount\fi.\the\theoremcount%
\xdef\theoremnum{\ifappend\applett\else\the\sectioncount\fi.%
\the\theoremcount}\fi\xdef\@currentlabel{\theoremnum}}

\def\allcorolnums{%
\ifspecialnumon\global\specialnumonfalse
\ifadded\global\addedfalse
\iffirstadded\global\firstaddedfalse
\global\advance\corolcount by1 \fi
\the\corolcount\addedlett%
\else$\rm\spnum$\def\corolnum{{$\rm\spnum$}}\fi%
\else\global\advance\corolcount by1 
\the\corolcount\fi}

%% use for Theorem, Corollary, Lemma, Proposition, Demonstration and similar.

\newcount\corolcount
\def\xcorol{Corollary}
\def\xtheorem{Theorem}
\def\xmaintheorem{Main Theorem}

\newif\ifthtitle
\def\proclaimtitle#1{\gdef\thethtitle{#1}\global\thtitletrue}


\let\saverparen)
\let\savelparen(
\def\rmparenl{{\rm(}}
\def\rmparenr{{\rm\/)}}
{
\catcode`(=13
\catcode`)=13
\gdef\makeparensRM{\catcode`(=13 \catcode`)=13
\let(=\rmparenl \let)=\rmparenr
\everymath{\let(\savelparen
\let)\saverparen}
\everydisplay{\let(\savelparen
\let)\saverparen}}
}

\def\medskip{\vskip\medskipamount}
\medskipamount=8pt plus.1\baselineskip minus.05\baselineskip
\def\proclaim#1{\vskip-\lastskip
\def\one{#1}\ifx\one\xtheorem\global\corolcount=0\fi
\ifsection\global\sectionfalse\vskip-6pt\fi
\medskip
{\elevensc#1}%
\ifx\one\xmaintheorem\global\corolcount=0
\gdef\theoremnum{Main Theorem}\else%
\ifx\one\xcorol\ \allcorolnums\else\ \alltheoremnums\fi\fi%
\ifthtitle\ \global\thtitlefalse{\rm(\thethtitle)}\fi.%
\hskip.5pc\bgroup\makeparensRM\it\ignorespaces}


\def\nonumproclaim#1{\vskip-\lastskip
\def\one{#1}\ifx\one\xtheorem\global\corolcount=0\fi
\ifsection\global\sectionfalse\vskip-6pt\fi
\medskip
{\elevensc#1}.\ifx\one\xmaintheorem\global\corolcount=0
\gdef\theoremnum{Main Theorem}\fi\hskip.5pc%
\bgroup\it\makeparensRM\ignorespaces}

\def\endproclaim{\egroup\medskip}

%% Use demo for Proof, Proof of, Definition, Example,
%% Remark, Case, Subcase, Conjecture, Note, Notation,
%% Convention, Construction and Step.
%% Any other use for demo will format similar to `Proof.'

\def\xproof{Proof}
\def\xremark{Remark}
\def\xcase{Case}
\def\xsubcase{Subcase}
\def\xconjecture{Conjecture}
\def\xstep{Step}
\def\xof{of}


\def\deconstruct#1 #2 #3 #4 #5 @{\def\one{#1}\def\two{#2}\def\three{#3}%
\def\four{#4}%
\ifx\two\empty #1\else%
\ifx\one\xproof%
\ifx\two\xof%
  \ifx\three\xcorol Proof of Corollary \rm#4\else%
     \ifx\three\xtheorem Proof of Theorem \rm#4\else\xone\fi%
  \fi\fi%
\else\xone\fi\fi.}


\def\pickup#1 {\def\this{#1}%
\ifx\this\xproof\global\let\go\demoproof
\global\let\enddemo\endproof\else
\ifx\this\xremark\global\let\go\demoremark\else
\ifx\this\xcase\global\let\go\demostep\else
\ifx\this\xsubcase\global\let\go\demostep\else
\ifx\this\xconjecture\global\let\go\demostep\else
\ifx\this\xstep\global\let\go\demostep\else
\global\let\go\demoproof\fi\fi\fi\fi\fi\fi}

\def\demo#1{\vskip-\lastskip
\ifsection\global\sectionfalse\vskip-6pt\fi
\def\one{#1 }\def\two{#1*}%
\setbox0=\hbox{\expandafter\pickup\one}\expandafter\go\two}

\def\numbereddemo#1{\vskip-\lastskip
\ifsection\global\sectionfalse\vskip-6pt\fi
\def\two{#1*}%
\expandafter\demoremark\two}

\def\demoproof#1*{\medskip\def\xone{#1}
{\ignorespaces\it\expandafter\deconstruct\xone {} {} {} {} {} @%
\unskip\hskip6pt}\rm\ignorespaces}


\def\demoremark#1*{\medskip
{\it\ignorespaces#1\/} \alltheoremnums\unskip.\hskip1pc\rm\ignorespaces}

\def\demostep#1 #2*{\vskip4pt
{\it\ignorespaces#1\/} #2.\hskip1pc\rm\ignorespaces}

\def\enddemo{\medskip}


\def\endproof{\ifmathqed\global\mathqedfalse\else
\parfillskip=0pt~~\hfill\qed\fi\medskip
\global\parfillskip0pt plus 1fil\relax
\gdef\enddemo{\medskip}}

\def\qed{\vbox{\hrule\hbox{\vrule height6pt\hskip6pt\vrule}\hrule}}

%% Proof box to be used in a \proclaim{}...\endproclaim environment

\def\proofbox{\parfillskip=0pt~~\hfill\qed\vskip1sp\parfillskip=
0pt plus 1fil\relax}

%%%

\let\endsteps\endcases
\let\endconjectures\endcases

%%%%%%%%%%%%%%%%%%%%%
%% 10) Listing

\def\listitem#1{\vskip1sp\indent\llap{(#1)}\ }

%% To use with asterisks:
\def\asteritem#1{\vskip1sp\indent\llap{$#1$}}


%%%%%%%%%%%%%%%%%%%%%%
%% 11) Article and Journal Table of Contents

\def\articlecontents{
\vskip20pt\centerline{\bf Table of Contents}\everypar={}\vskip6pt
\bgroup \leftskip=3pc \parindent=-2pc 
\def\item##1{\vskip1sp\indent\hbox to2pc{##1.\hfill}}}

\def\endcontents{\vskip1sp\leftskip=0pt\egroup}

\def\journalcontents{\vfill\eject
\def\currannalsline{\hfill}
\global\titletrue
\vglue3.5pc
\centerline{\tensc\hskip12pt TABLE OF CONTENTS}\everypar={}\vskip30pt
\bgroup \leftskip=34pt \rightskip=-12pt \parindent=-22pt 
  \def\\ {\vskip1sp\noindent}
\def\pagenum##1{\unskip\parfillskip=0pt\dotfill##1\vskip1sp
\parfillskip=0pt plus 1fil\relax}
\def\name##1{{\tensc##1}}}

%% default values

\institution{}
\onpages{0}{0}
\def\thetitle{Title ???}
\def\theauthors{Authors ???}
\def\thereceived{}
\def\therevised{}


%% LaTeX Output

\newif\iftitle

\def\amheadline{\iftitle%
\hbox to\textwidth{\hss\currannalsline\hss}\else%
\hbox to\textwidth{\ifodd\c@page%
\hfill\thetitle\hfill\llap{\elevenrm\the\c@page}%
\else\rlap{\elevenrm\the\c@page}%
\hfill\theauthors\hfill\fi}\fi}


\def\ps@headings{\def\@oddfoot{\global\titlefalse}%
\def\@evenfoot{\global\titlefalse}%
\def\@evenhead{\amheadline}%
\def\@oddhead{\amheadline}}

\ps@headings \pagenumbering{arabic} \onecolumn 

\pagestyle{headings}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  Output routine to make back-to-back pages match.

\def\bindingoffset{\ifodd\c@page \hoffset=-18pt\else\hoffset0pc\fi}

\voffset=-72pt

\def\@outputpage{\bindingoffset\begingroup\catcode`\ =10 
     \let\-\@dischyph \let\'\@acci \let\`\@accii \let\=\@acciii
    \if@specialpage 
     \global\@specialpagefalse\@nameuse{ps@\@specialstyle}\fi
     \if@twoside 
       \ifodd\count\z@ \let\@thehead\@oddhead \let\@thefoot\@oddfoot
            \let\@themargin\oddsidemargin
          \else \let\@thehead\@evenhead
          \let\@thefoot\@evenfoot \let\@themargin\evensidemargin
     \fi\fi
     \shipout
     \vbox{\normalsize \baselineskip\z@ \lineskip\z@ 
           \let\par\@@par %% 15 Sep 87
           \vskip \topmargin \moveright\@themargin
           \vbox{\setbox\@tempboxa
                   \vbox to\headheight{\vfil \hbox to\textwidth
                                       {\let\label\@gobble \let\index\@gobble
                                         \@thehead}} %% 22 Feb 87
                 \dp\@tempboxa\z@
                 \box\@tempboxa
                 \vskip \headsep
                 \box\@outputbox
                 \baselineskip\footskip
                 \hbox to\textwidth{\let\label\@gobble 
                           \let\index\@gobble  %% 22 Feb 87
                           \@thefoot}}}\global\@colht\textheight
           \endgroup\stepcounter{page}\let\firstmark\botmark}


\endinput

